Question

Match the LIST-I with LIST-II

LIST-I Matrix \(A\)	LIST-II \(|\text{adj}\,A|\)
A. \(A=\begin{bmatrix}2&4\\1&3\end{bmatrix}\)	I. 1
B. \(A=\begin{bmatrix}5&2\\7&4\end{bmatrix}\)	II. 7
C. \(A=\begin{bmatrix}1&0\\0&1\end{bmatrix}\)	III. 2
D. \(A=\begin{bmatrix}6&1\\5&2\end{bmatrix}\)	IV. 6

Choose the correct answer from the options given below:

• A. A - I, B - II, C - III, D - IV
• B. A - IV, B - II, C - III, D - I
• C. A - III, B - IV, C - I, D - II
• D. A - III, B - IV, C - II, D - I

Hint:

For a 2 by 2 matrix, the determinant of the adjoint equals the determinant of the matrix itself.

Answer 1

The Correct Option is C

Step 1: Idea:
Write the adjoint of each 2 by 2 matrix directly and take its determinant. For $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ the adjoint is $\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$.

Step 2: Matrix A.
$\text{adj}\,A=\begin{bmatrix}3&-4\\-1&2\end{bmatrix}$. Its determinant is $3\cdot 2-(-4)(-1)=6-4=2$. This is value III.

Step 3: Matrix B.
$\text{adj}\,B=\begin{bmatrix}4&-2\\-7&5\end{bmatrix}$. Its determinant is $20-14=6$. This is value IV.

Step 4: Matrix C.
The adjoint of the identity matrix is the identity matrix, with determinant 1. This is value I.

Step 5: Matrix D.
$\text{adj}\,D=\begin{bmatrix}2&-1\\-5&6\end{bmatrix}$. Its determinant is $12-5=7$. This is value II.

Step 6: Read the option.
The pairs A-III, B-IV, C-I, D-II appear in option 3 only.

Final Answer:
The correct match is A - III, B - IV, C - I, D - II. \[ \boxed{\text{A-III, B-IV, C-I, D-II}} \]